1
Quantum Political Economics
Erman ZENG1
Abstract: The mathematical characterization of “the Productive Force” of a macroeconomic system is based on Newton mechanics and quantum physics, which isexpressed as the product of the growth rate of the profit rate (p) and the surplus value (M),showing several quantum qualities similar to the photon. The one-dimensional linearharmonic oscillator model can correlate the angular frequency with the change rate of therate of profit thus with the economic growth rate, leading to the quantum-like interpretationof various economic cycles. The matrix mechanics analysis of the Leontief’s input-output
table gives the Schrodinger-like value-price transformation eigen-equation ( g ),
with the reduced organic composite of capital as the eigenvalue of the price wave function,leading to the "two Cambridge controversy" resolved.Key words: quantum mechanics, harmonic oscillator, econophysics, labor theory of value,matrix operator, eigen-vector, eigen-function, eigenvalue, general equilibrium, wavefunction, transformation problem, input-output,Two Cambridge ControversyJEL:A12 E11 E25 C67 C68 D24 D46 D57 D58 E25 O47
1、Classic mechanics modelSimilar to the fractal space-time theory2, the natural logarithm of the price of a
commodity (lnP) is defined as the "displacement" (q) of the classical Newtonianphysics, the speed (v=dq/dt) corresponding to the rate of the profit (p’):
PdtdP
dtPdp
ln'
the momentum (mv) as the surplus value:M =p’Cv,Cv=C+V, C=nK:constant capital,n:turnover rate of capital K; V=wL:variable capital,w:wage rate,L:labour; thus the“productive forces3”(Φ) can be expressed as:
KLwnn
gg
CvCnnpnKpCM
nn
KCvVKwhenKnKKVnVCCv
mYgyLYgFYMpdtpdpCvp
dtdpCv
/1,M''
;)1(pM)1(
,p'M,0nKC0n:,)1()(
*)1(*)1(''''
00
00
0
,,
1 Amoy Institute of Technovation(XiaMen Productivity Promotion Center),1300 JiMei Blvd., XiaMen, FuJian
361024, PRChina, Email: [email protected] Nottale, L. (2011)Scale Relativity and Fractal Space-Time: a NewApproach to Unifying Relativity and QuantumMechanics, Imperial College Press.3 曾尔曼,《厦门科技》4(2016)50-56.
2
Formula Φ=Φ0(n+1) reflecting the photon-like quantum character, the productivityforces(Φ) includes all the elements described by Marx4 such as the level of divisionof labor (1-), labor productivity y, number of workers L showing quantum charactertoo, the organic composition of capital g (OCC, *: general equilibrium), theproductivity coefficient F, and the Solow residual m (technical progress rate), underthe condition of the Cobb-Douglas production function5: m=F, other analogies are asfollows (Table 1):
From the definition of annual profit rate:
)2
1(')!2
(')1!
(')1('ln
'ln'
02
2000
0
tptptiptppt
ppt
ip
ppe
ppP
epdtPdp
ii
i
ii
pt
pt
prices tend to increase, especially when p is positive; and the quantum character isdisplayed obviously。
The least action principle can be used to determine the economic equilibrium g*:
222
2
'22
''22
)'('''
''''
)(ln
pCvMpMpCvMV
vpVTL
pCvMpdtpdpM
dtpdppCv
dtdpCv
MpP
vmmaqVF
4 Karl Marx, The German Ideology, 18455 Cobb C.W., Douglas P.H. “A Theory of Production”, Amer. Econ. Rev. 1928,8(1), Spp1.139-165.6 曾尔曼,(2016)《马克思生产力经济学导引》,厦门大学出版社.(Erman ZENG,“Introducing MarxianEconomics of Productive Forces”,Xiamen Univ. Press,2016)
Table 1.Classic
MechanicsPoliticalEconomics6
mass m Cv=C+V Capital input
displacement q lnP P:price
velocity v=dq/dt p'=dlnP/dt Profit rate
acceleration a=dv/dt dp'/dt Growing rate of p’
Momentum p=mv M=p’Cv Surplus value
potential E T=p^2/2m Mp’=Ψ potential
force F=ma Φ=Mp Productive forces
3
/*,;0
)1(0
)]'2
('
[)'
(
'ln,'''
''22ln
2
VgpMCdCv
CgCVgvCCYVgpMp
MpdtdMpCv
pdtd
pL
dtd
pPdtdP
dtPdMpMp
dtdpCv
tpppCv
PL
;
According to the Lagrange equation, the "quality"of a system thus the capital inputand output (Cv=C+V) is conserved. The least action principle reveals that thegrowth rate of constant capital C equals to the growth rate of the reduced organiccomposition of capital β, the growth rate of surplus value M is equal to the growthrate of profit rate P, thus the change rate of the capital output coefficient of the CDproduction function .
In addition, according to the Noether theorem, it is shown that under thecondition of no external force, the conservation of energy is the conservation of thevalue of the commodity as well as the profit rate, unless the division of labor ischanged:
0000
)()(0'0
2/'32/'3)2/'(' 2
dQdCvvCQdCvCvdQCvQ
dtd
CvM
dtd
dtdp
dtdH
CvpMpMpMpLqpH
;
1:,1:
,)1()1(1)]1(1)[1('
1'''1'1
LR
RL
LR
ddddddp
pPppQYC
dL:division of labor,dR:roundabout production.
2、Harmonic Oscillator ModelOne dimensional linear harmonic oscillator7 model can be used to approximate
an equilibrium state of a system, the potential energy and kinetic energy are equal:
7 曾谨严,(2016,第 VI 版)《量子力学》,科学出版社.
4
RR
RR
R
RL
RRR
t
t
dddndnCnnpM
dnMnpnKpCMMp
ggdd
mddggpmdmp
KLwnn
wLnKnK
CvC
pFm
pepggdtpdp
kCvM
kMp
kKpn
kCp
kM
kMpPeP
TN
Mkp
Ck
Vk
gVk
Cvk
CvpCvkYp
T
YYM
pPdtPdp
pP
MPkpPkMp
Ntt
tWptWPCvk
PpXPmCvPXp
pCvMpHXmkXE
Cvmkkxxmamf
11
;1
'')*)((
;',',''1
/
;
'ln'ln,)1(
**1'ln
'''lnln
;4
)41(')1(1
/
111;
;0':when1or0)ln(ln0
'lnlnln
;4
,1tg
cos',sinlnln',ln,,ln'
'2
'21,)(
21
21
,
00
0
0
22
2
0
220
2
22000
;
,
)(:,
;
N is a natural number, therefore, the productive forces , “price coefficient”lnP,therate of technological progress m, various types of capitals (C, V, M) and the organiccomposition of capital g, the growth rate of profit rate (p')p , the labor divisioncoefficient dL and the production roundaboutness dR, all the parameters of theeconomic system show the quantum characters . Similar to the quantum mechanics ofthe photon quantized energy E=h and the one-dimensional harmonic oscillatorenergy En=(n+1/2)h /2 , the economic system directly reflected its quantum-likebehavior in the analytic expression of the "productive forces" =(n+1)M0;every
5
intermediate inputs (C) and output surplus value (M) are right the contribution to theproductive forces’ growth, not only in a micro commodity, but also a meso industry,and even the whole macro economy.
The development of productive forces mainly resulted from the increased ratesof both the capital turnover (n) and the production roundaboutness dR , which canreduce the capital investment (C), avoid the falling rate of profit; in other words, theacceleration of the capital turnover can offset the insufficient capital investment andlower profit rate.
Table 2 shows the calculation of the economic business cycles based on variousgrowth rates of GDP (≈p):
Table 2 t= /4*
p= (month) year (1+4N)
0.255 47.1 3.9 50.141 84.8 7.1 90.098 122.5 10.2 130.075 160.2 13.4 170.061 197.9 16.5 210.051 235.6 19.6 250.039 311.0 25.9 330.021 574.9 47.9 610.017 725.7 60.5 770.015 801.1 66.8 850.014 838.8 69.9 890.013 951.9 79.3 1010.011 1065.0 88.8 1130.010 1178.1 98.2 1250.009 1328.9 110.7 141
After a comprehensive analysis of different economic cycles based upon hisinnovation theory, in 1936 Schumpeter8 proposed that each long cycle(the Kondratiev wave or long technological cycle of 45 to 60 years) includes 6medium cycles (the Juglar fixed-investment cycle of 7 to 11 years and the Kuznetsinfrastructural investment cycle of 15 to 25 years), each cycle includes three shortcycle (the Kitchin inventory cycle of 3 to 5 years).
Since the mid eighteenth Century, the industrial revolution has occurred threetimes. The first created the steam machine era (1760-1840, 80years), marking thetransition from agricultural civilization to industrial civilization; the second industrialrevolution moved the human being into the electric chemical age (1840-1950,110years); after the World War II, the third industrial revolution began and created anelectronic-information age (1950-2011, 61years). Since 2011, the era of artificial
8 Schumpeter, J. A. (1954). History of Economic Analysis. London: George Allen & Unwin.
6
intelligence stared . Each economic cycle with different length of the year can beexplained properly.
3. Economic Operator--Matrices MechanicsAccording to the function system of Marx's theory of value9, all the input-output
relationships are as follows:
MY
pP
MQVCebMueFunctionSurplusVal
FppwnmTheoremOkishio
LKeAVCeaVMYnctionoductionFuMarxVC
CggreducedOCC
VCgiteanicComposCapitalOrg
VCeBCQTheoryLaborValue
pt
mtFt
ft
,'',
;)1(:""
,Pr1
,1
;YCMV
10
10
10
10
:
:
;::
:
''
,''
,''1)'1)((
pPMQ
QC
MC
pPYM
MQ
QC
YC
PppVCC
MVCC
QC
For Marx’s two Departments of production system, the Leontief intermediateinput nXn matrix is non-negative, according to the Perron-Frobenius theorem10 thereis always a positive eigenvalue , and {Pi} standing for the Marx price eigenvector:
1
1
2
2
2
2
1
1
2
2
1
121
2
2
1
2
1
2
2
2
2
1
1
1
1
212121
2
1
2
1
2
1
22
11
222221
111211
2121222222
2121111111
1,1
1,
''
''
''
''
:
''''''
1,'00'
''''
''''''
)Re:(,
QY
QC
QY
QC
QY
QC
ppp
pp
QY
QC
QY
QC
or
CYQQYCP
ppp
pp
YCYC
QpYpCpQpYpCp
QQQYYYQYCMVCproductionSimpleCYCCCQYCMVC
ii
ii
;
;
;
9 曾尔曼,(2016)《马克思生产力经济学导引》,p122,厦门大学出版社.(Erman ZENG,“Introducing MarxianEconomics of Productive Forces”,Xiamen Univ. Press,2016)10 Seneta, E. (1973) Non-negative Matrices – An Introduction to Theory and Applications. London: George Allenand Unwin.
7
112
212
21212
2121
22222
22
2112
211
12121
11
112
121
111
222
2222
2121
2
111
1212
1111
1
1
2
2
1112
2
2
1
12
2
22121
2
1
2
1
2
1
2
2
2
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1
1
1
1
')(
-'
)(')(''')''()''(
;-ln-)(
-'
)('')''('')''(
''''''
;''''''''
''',1:1
''
'')2
''',1:1)1
1,1
1
1001
1
1
''
''
''
''
YdtppppdC
dtdp
dtdpCpp
dtdC
dtdpCp
dtdC
dtdpYQp
dtYQd
ppdtd
dtppppdY
ppdtdY
dtdp
dtdpY
dtdpCQ
dtdpYp
dtdYp
dtQCd
dtdpQp
dtdQ
dtdpYp
dtdY
dtdpCp
dtdC
dtdpQp
dtdQ
dtdpYp
dtdY
dtdpCp
dtdC
CQCpp
QY
QC
CYQpp
Pp
pp
P
P
P
P
QY
QC
QY
QC
ii
)(
)()(
)(
;;
Based on the conservation of the value of input-outputs, the static/dynamicequilibrium conditions of Marx's simple reproduction/complex reproduction areobtained emergently. Or in a three-sector economic system--production material,living material, capital goods-- as proposed by J. Winternitz11, we also have:
ii Pp
ppp
ppp
ppp
QM
QV
QC
QM
QV
QC
QM
QV
QC
VMCCMVYQQQMVCorPPP
Q
PPP
MVCMVCMVC
MMMPMPQPMPVPCPVVVPVPQPMPVPCPCCCPCPQPMPVPCP
1,100010001
''
''
''
''
''
''
''
''
''
),(,:
,00
0000
)(:)3()(:)2()(:)1(
3
2
1
3
2
1
3
2
1
3
3
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
3232111321321
3
2
1
3
2
1
3
2
1
333
222
111
3213333333231
3212222232221
3211111131211
All Marx-Leontief12 input and output functions have the same form, not matter in thevalue form or in the price form:
11 Winternitz,J. ‘Values and Prices: a solution to the so-called transformation problem’, Econ. Jour. 1948, 58,276-280.12 Leontief,W. “Input-Output Economics”. 2nd ed., New York: Oxford University Press, 1986.
8
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''1,)''(''
'
;''1,'
''
3
1
323
1 32
3
1
3
1
3
1
3
1
i
ii
YY
i
ii
Y
iii
Y
i i
i
i i
i
Q
ii
Q
YMpVp
Pp
PM
PV
P
MV
PY
PQQ
Pp
PQ
P
Q
PQ
IPp
CvIpM
VCMr
PPIIICvdCvVCcVCc
PPIvC
IpbbVCebM
IpaBVCeaY
IpBBVCeBVCeB
PPP
QPV
PCeBQ
PQ
M
M
VCVC
M
pt
Y
Ft
Q
ftft
VC
Q
VC
ft
Q
'''
''
)1ln(,'''''
;','''';',''''
;','''''')')'('
110
101
00
10
00
10
00
10
101
10
Since the mechanics quantities can be characterized as operators, the economicinput-output eigen-equations are:
nctioniceEigenfuCCPPC
CPPPQ
CCC
PPpPQC
PQ
PQ
PPC
PQC
jiji
ijj i
ij
ii
i
ji
i
j i
ij
iii
i
jj i
ij
i
i
i
i
i
i
j j
ij
Pr:)''(11''1
'1)
''(
''
11'1
1''''
''
'
ijj i
ij
ii
i
jj i
i
i
ij
iii
i
jj i
ij
i
i
i
ii
i
i
i
jiji
i
i
j j
ij
PPCC
PpPYC
CC
PPpPYC
PY
PMV
pP
CQ
pPC
pYC
11''1
'1
''
''11
'1
''
''''
''
''
'
C’ij/C’i is the Leontief intermediate input coefficient matrix, all the eigen equations,eigenvalues and eigenvector of the price coefficient are obtained; similarly:
iii
i
jj i
ij
i
i
i
i
j j
ij
PPpPMC
PM
pPC
pMC 11
'1
'''
''
'
ii
jj ii
ij
iii
i
jj i
ij
i
i
i
i
j j
ij
PPMQC
MQC
VCCor
PPPVC
PV
PC
gVC
11''
',:
;111
1'''
1'
1
The eigenvalue (β/P’) of the eigen-equation of the intermediate input C and the total
9
output of Q demonstrated that there existed in the economic system a ratio of thereduced organic composition of capital (ROCC) and the rate of profit of P' (P '=p' +1) ,rather than the "uniform profit rate p’"13; Marx's "uniform profit rate" in the "DasKapital" should be ROCC (β), which is the eigenvalue of the eigen-equation of theintermediate input C and the total input cost Cv=C+V。
The ratio (β/p’) is the eigenvalue of the eigen-equation of the input--constantcapital C and the output--surplus value M;the ratio ( /p’) is the eigenvalue of theeigen-quation of the intermediate input C and the end use Y.Details are as following:
jjii
ij
nnn
jjii
ij
nnn
jji
ij
nn
jji
ij
nn
jji
ij
nn
jji
ij
nnnnnnn
n
n
ppMQ
C
p
pp
MQ
MQMQ
ppVC
C
p
pp
VC
VCVC
ppVC
p
pp
V
VV
pp
pYC
p
pp
Y
YY
p
pp
pMC
p
pp
M
MM
p
pP
pQC
p
pp
Q
pp
pp
CCC
CCCCCC
'''
''
''''
;''
'
''
''''
;1'
'
'
''
1
;''
'
'
''
'
;''
'
'
''
'
;''
'
'
''
'1'''
''''''
2
1
22
11
2
1
22
11
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
21
22221
11211
Wave function is the price eigen-vector:
1*
,2)()(
)(
i
ij
ij
i
jii
ppp
p
determined by price eigen-function:
13 王璐,柳欣,(2006)《马克思经济学与古典一般均衡理论》,人民出版社.
10
kiki
nnnnnnn
n
n
CC
p
pp
C
CC
p
pp
CCC
CCCCCC
'',
'
''
'''
''''''
2
1
2
1
2
1
21
22221
11211
Similar to Schrodinger function, the price wave function can be defined as:
iiji
ii
ij
ii
ij CCMQ
CVC
CgorROCCoperat
g
'',''
'''
'ˆ:
;ˆ
Similar to the intermediate input C=nK, each output--the added value Y, the surplusvalue of M, the variable capital/human capital V--displayed the quantum nature too.According to the USA 1997-2014 input-output data14, the eigenvalue (=C’/Q’) ofintermediate input C-total output Q is about 0.479 (Table 3),however in the year of1997-2002、2004、2007、2008, the price eigenvectors are negative:
In addition, the eigen-equations can also be expressed in Dirac ket-bra notation15:
14 http://www.bea.gov/industry/io_annual.htm15 PAM Dirac (1939). "A new notation for quantum mechanics".Mathematical Proceedings of the CambridgePhilosophical Society. 35 (3): 416–418;狄拉克,(1979)《量子力学原理》,科学出版社.
11
yp
wpY
Vp
QpCPCCQQP
MQCvVp
Mp
Yp
QCPCPPCP
i jjij
1'
1'
'1'ˆ,/'
)''('1'
''
''
'1''ˆ
1
11
iinnn
nnn
n
nnnnn
n
n
n
CCCCCPCPCP
CpC
CpCCpC
PPP
p
pp
CCC
CCCCCC
PPP
''''
'
''
,,
'''
''''''
***
212211
222
111
21
2
1
21
22221
11211
21
Because the production and sale of each commodity is affected by other commodities,it is impossible to solve the "price wave function" independently like the schrodingerequation. Because all the commodities’ price are known, using the Leontiefintermediate input n x n matrix, the price eigenstate vector as the wave function areobtained. "Two Cambridge Controversy"16 about the difficulty of the aggregation ofheterogeneous capital goods in the CD production function, the “uniform profit rate”in the general equilibrium state problem, and the wage income distribution17 issueswere resolved. The wage (w) is proportional to the labor productivity (y) andinversely proportional to the rate of profit p’.
4. DiscussionThe discreteness of physical goods in the market, and the price based currencies
have the smallest unit (¥ ; $cent, penny, etc.) and the individual nature of the laborforce, is the objective basis of the discussion of the quantum nature of macroscopiceconomic system。 It is worth mentioning that the uncertainty principle in quantummechanics discovered by Heisenberg18 90 years ago, reflects that the value and theprice cannot be determined simultaneously:
16 Joseph E. Stiglitz, "The Cambridge-Cambridge Controversy in the Theory of Capital; AView from New Haven:A Review Article," Journal of Political Economy, 82(4), Jul.-Aug. 1974: 893-903.17 Erman Zeng,"From Labor Theory of Value to Price Eigen Function--Microfoundation of General Equilibrium",http://vixra.org/abs/1702.017418 Heisenberg, W. The physical content of quantum kinematics and mechanics. In Wheeler, J. A. & Zurek, W. H.(eds.) Quantum Theory and Measurement, 62–84 (Princeton UP, Princeton, NJ, 1983). [Originally published: Z.Phys., 43, 172-98 (1927)].
12
2/)(ln)'(1'/',/
2/)(ln],[ln],[1
QPppPQMMPMPipx ,
The negative value of the price eigen-vectors, corresponding to USA’s economic /financial crisis (2002/2008), need further empirical research, which may be carriedout by the Feynman path integral19 of quantum mechanics representation, to study thetime evolution the price state vector in the macro-economic system. It can provide thegreat help for the decision-making.
19 Feynman, R. P.; Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. New York: McGraw-Hill.